In this section, we will derive the explicit closed-form expression of the ergodic capacity with imperfect CSI. First of all, the ergodic achievable rate can be expressed by [1]}
\(\mathbb {E}\left\lbrace  R \right\rbrace  = \mathbb {E}\left\lbrace  {{{\log }_2}\left( {1 + \text{SNR}} \right)} \right\rbrace  \le {\log _2}\left( {1 + \mathbb {E}\left\lbrace  {\text{SNR}} \right\rbrace } \right),\) 
